Subgroups — #10711544

Order 768.

314 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#364	C₂	1	1	yes	maximal, characteristic
2	C₄×#815	C₂	1	1	yes	maximal, characteristic
3	C₂×#4463	C₂	1	1	yes	maximal, characteristic
4	#67812	C₂	1	2	yes	maximal
5	#67811	C₂	1	2	yes	maximal
6	#12882	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₃₂	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#815	C₄	1	1	yes	characteristic
10	C₂×#815	C₄	1	1	yes	characteristic
11	C₂×#815	C₂²	1	1	yes	characteristic
12	C₂×#815	C₂²	1	1	yes	characteristic
13	C₂×#812	C₂²	1	2	yes	—
14	C₄²×Dic₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₃⋊C₈	C₄	1	1	yes	characteristic
16	C₃×#364	—	2	2	no	—
17	#4463	—	2	2	no	—
18	C₂×#364	—	3	1	no	—
19	C₂×#364	D₃	1	1	yes	characteristic
20	C₄×#97	—	3	1	no	—
21	#1377	—	3	2	no	—
22	#1375	—	3	2	no	—
23	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
25	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₃	—	2	2	no	—
27	C₂×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₃×M₃₂	—	2	2	no	—
32	C₃×M₃₂	—	2	2	no	—
33	C₂×C₃×C₁₆	—	2	2	no	—
34	C₃×C₄×C₈	D₄	1	2	yes	—
35	C₃×C₄×C₈	Q₈	1	2	yes	—
36	#815	—	2	2	no	—
37	#815	—	2	1	no	—
38	#815	—	2	1	no	—
39	#815	—	2	1	no	—
40	#815	—	2	1	no	—
41	#815	C₈	1	4	yes	—
42	#812	—	2	2	no	—
43	#812	—	2	2	no	—
44	C₂×C₃⋊C₁₆	—	2	2	no	—
45	C₄×C₃⋊C₈	—	2	2	no	—
46	C₂×M₃₂	—	3	2	no	—
47	C₂×M₃₂	C₂×D₃	1	2	yes	—
48	C₂×#97	—	3	1	no	—
49	C₂×#97	—	3	1	no	—
50	C₂×#97	—	3	1	no	—
51	C₂×#97	—	3	1	no	—
52	C₂×C₄×C₈	—	3	1	no	—
53	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
54	C₄³	—	3	1	no	—
55	#364	—	2	2	no	—
56	#364	—	6	2	no	—
57	C₂²×C₃×C₄	—	2	1	no	—
58	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂²×Dic₃	—	2	1	no	—
60	C₂²×Dic₃	M₁₆	1	2	yes	—
61	C₂×C₃×C₈	C₄⋊C₄	1	2	yes	—
62	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
63	C₂×C₃×C₈	M₁₆	1	2	yes	—
64	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₃×C₈	#41	1	1	yes	characteristic
67	C₂×C₃×C₈	#41	1	1	yes	characteristic
68	C₂×C₃×C₈	#41	1	1	yes	characteristic
69	C₂×C₃×C₈	#41	1	1	yes	characteristic
70	C₃×C₄²	—	2	1	no	—
71	C₃×C₄²	C₄⋊C₄	1	2	yes	—
72	C₄×Dic₃	—	2	1	no	—
73	C₄×Dic₃	—	2	1	no	—
74	C₄×Dic₃	—	2	1	no	—
75	C₄×Dic₃	—	4	4	no	—
76	C₄×Dic₃	C₂×C₈	1	2	yes	—
77	C₂×C₃⋊C₈	—	2	1	no	—
78	C₂×C₃⋊C₈	—	2	1	no	—
79	C₂×C₃⋊C₈	—	2	1	no	—
80	C₂×C₃⋊C₈	—	2	1	no	—
81	C₂×C₃⋊C₈	C₂×C₈	1	2	yes	—
82	C₂×C₃⋊C₈	M₁₆	1	2	yes	—
83	C₃×C₁₆	—	4	2	no	—
84	C₃⋊C₁₆	—	4	2	no	—
85	C₂²×C₈	—	3	1	no	—
86	C₂²×C₈	—	3	1	no	—
87	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
88	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
89	C₂×C₄²	—	3	1	no	—
90	C₂×C₄²	—	3	1	no	—
91	C₂×C₄²	—	6	2	no	—
92	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
93	M₃₂	—	2	2	no	—
94	M₃₂	—	2	2	no	—
95	M₃₂	—	6	2	no	—
96	M₃₂	—	6	2	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	6	2	no	—
99	#97	—	3	4	no	—
100	#97	—	6	2	no	—
101	#97	—	6	1	no	—
102	#97	—	6	1	no	—
103	#97	—	6	1	no	—
104	#97	—	6	1	no	—
105	C₄×C₈	—	6	2	no	—
106	C₄×C₈	Dic₆	1	2	yes	—
107	C₄×C₈	#67	1	2	yes	—
108	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₃×C₄	—	2	4	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₃×C₄	#117	1	1	yes	characteristic
113	C₂×C₃×C₄	#117	1	1	yes	characteristic
114	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
117	C₂×Dic₃	—	2	2	no	—
118	C₂×Dic₃	—	2	2	no	—
119	C₂×Dic₃	—	4	2	no	—
120	C₂×Dic₃	—	4	2	no	—
121	C₃×C₈	C₄⋊C₈	1	4	yes	—
122	C₃×C₈	#124	1	1	yes	characteristic
123	C₃×C₈	#124	1	1	yes	characteristic
124	C₃×C₈	#125	1	1	yes	characteristic
125	C₃×C₈	#125	1	1	yes	characteristic
126	C₃⋊C₈	—	2	2	no	—
127	C₃⋊C₈	—	2	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	3	2	no	—
130	C₂²×C₄	—	6	1	no	—
131	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
132	C₂×C₈	—	3	2	no	—
133	C₂×C₈	—	3	2	no	—
134	C₂×C₈	—	6	1	no	—
135	C₂×C₈	—	6	1	no	—
136	C₂×C₈	—	6	1	no	—
137	C₂×C₈	—	6	1	no	—
138	C₂×C₈	D₃×C₈	1	2	yes	—
139	C₂×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
140	C₂×C₈	Dic₃⋊C₄	1	2	yes	—
141	C₂×C₈	C₂×Dic₆	1	1	yes	characteristic
142	C₂×C₈	#237	1	1	yes	characteristic
143	C₂×C₈	#237	1	1	yes	characteristic
144	C₂×C₈	#239	1	1	yes	characteristic
145	C₂×C₈	#239	1	1	yes	characteristic
146	C₂×C₈	C₂×#67	1	1	yes	characteristic
147	C₄²	—	2	1	no	—
148	C₄²	—	3	2	no	—
149	C₄²	—	6	1	no	—
150	C₄²	—	6	1	no	—
151	C₄²	—	6	1	no	—
152	C₄²	—	12	4	no	—
153	C₄²	Dic₃⋊C₄	1	2	yes	—
154	C₁₆	—	4	2	no	—
155	C₁₆	—	12	2	no	—
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	#423	1	1	yes	characteristic
158	C₂²×C₃	#431	1	1	yes	characteristic
159	C₂²×C₃	#432	1	1	yes	characteristic
160	C₃×C₄	—	2	4	no	—
161	C₃×C₄	#382	1	1	yes	characteristic
162	C₃×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
163	C₃×C₄	#431	1	1	yes	characteristic
164	C₃×C₄	#432	1	1	yes	characteristic
165	Dic₃	—	4	2	no	—
166	Dic₃	—	4	2	no	—
167	C₂³	D₃×M₁₆	1	1	yes	characteristic
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	6	2	no	—
171	C₂×C₄	—	6	2	no	—
172	C₂×C₄	—	12	2	no	—
173	C₂×C₄	—	12	2	no	—
174	C₂×C₄	#892	1	1	yes	characteristic
175	C₂×C₄	#892	1	1	yes	characteristic
176	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
177	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
178	C₂×C₄	D₃×M₁₆	1	1	yes	characteristic
179	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
180	C₈	—	6	2	no	—
181	C₈	—	6	2	no	—
182	C₈	Dic₃⋊C₈	1	4	yes	—
183	C₈	#890	1	1	yes	characteristic
184	C₈	#890	1	1	yes	characteristic
185	C₈	#936	1	1	yes	characteristic
186	C₈	#947	1	1	yes	characteristic
187	C₂×C₃	—	2	1	no	—
188	C₂×C₃	—	2	1	no	—
189	C₂×C₃	#1831	1	1	yes	characteristic
190	C₂×C₃	#2165	1	2	yes	—
191	C₂²	—	2	2	no	—
192	C₂²	#4833	1	1	yes	characteristic
193	C₂²	#4835	1	1	yes	characteristic
194	C₂²	#5118	1	1	yes	characteristic
195	C₄	—	2	4	no	—
196	C₄	—	12	2	no	—
197	C₄	—	12	2	no	—
198	C₄	#4833	1	1	yes	characteristic
199	C₄	#4835	1	1	yes	characteristic
200	C₄	#4977	1	1	yes	characteristic
201	C₄	C₂×Dic₃⋊C₈	1	1	yes	characteristic
202	C₃	#12882	1	1	yes	characteristic, complemented, Sylow
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	#69298	1	2	yes	—
206	C₂	#70443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#892
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₃
Fitting	C₂×C₃×#364	C₂
socle	C₂²×C₃	#432
prime radical	C₂³×C₃	C₂×M₁₆
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×#364	D₃
hypocenter	C₃	#12882
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#364

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
202. C₃	#12882	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#364	2
C₄×#815	2
C₂×#4463	2
#67812	2
#67811	2
#12882	3